Question

We can identify the set V of all 3×3-matrices (real coefficients) with vector space R9. Show that the set of all 3 × 3 symmetric matrices is a vector subspace of V .

Answer #1

Determine if the given set V is a subspace of the vector space
W, where
a) V={polynomials of degree at most n with p(0)=0} and W=
{polynomials of degree at most n}
b) V={all diagonal n x n matrices with real entries} and W=all n
x n matrices with real entries
*Can you please show each step and little bit of an explanation
on how you got the answer, struggling to learn this concept?*

We have learned that we can consider spaces of matrices,
polynomials or functions as vector spaces. For the following
examples, use the definition of subspace to determine whether the
set in question is a subspace or not (for the given vector space),
and why.
1. The set M1 of 2×2 matrices with real entries such that all
entries of their diagonal are equal. That is, all 2 × 2 matrices of
the form: A = a b c a
2....

If V is a vector space of polynomials of degree n with real
numbers as coefficients, over R, and W is generated by
the polynomials (x 3 + 2x 2 − 2x + 1, x3 + 3x 2 − x + 4, 2x 3 +
x 2 − 7x − 7),
then is W a subspace of V , and if so, determine its basis.

Consider W as the set of all skew-symmetric matrices of size
3×3. Is it a vector space? If yes, then ﬁnd its dimension and a
basis.

Prove that the singleton set {0} is a vector subspace of the space
P4(R) of all polynomials of degree at most 3 with real
coefficients.

Let V be the vector space of 2 × 2 matrices over R, let <A,
B>= tr(ABT ) be an inner product on V , and let U ⊆ V
be the subspace of symmetric 2 × 2 matrices. Compute the orthogonal
projection of the matrix A = (1 2
3 4)
on U, and compute the minimal distance between A and an element
of U.
Hint: Use the basis 1 0 0 0
0 0 0 1
0 1...

Determine whether the given set ?S is a subspace of the vector
space ?V.
A. ?=?2V=P2, and ?S is the subset of ?2P2
consisting of all polynomials of the form
?(?)=?2+?.p(x)=x2+c.
B. ?=?5(?)V=C5(I), and ?S is the subset of ?V
consisting of those functions satisfying the differential equation
?(5)=0.y(5)=0.
C. ?V is the vector space of all real-valued
functions defined on the interval [?,?][a,b], and ?S is the subset
of ?V consisting of those functions satisfying
?(?)=?(?).f(a)=f(b).
D. ?=?3(?)V=C3(I), and...

What is the highest possible dimension of a subspace of M_n
(R) (set of n×n matrices with real coefficients with its usual
vector space structure) that only contains invertible matrices (and
0) ?

9. Let S and T be two subspaces of some vector space V.
(b) Define S + T to be the
subset of V whose elements have the form (an
element of S) + (an element of
T). Prove that S +
T is a subspace of V.
(c) Suppose {v1, . . . ,
vi} is a basis for the
intersection S ∩ T. Extend this with
{s1, . . . ,
sj} to a basis for
S, and...

Let P2 denote the vector space of polynomials in x with real
coefficients having degree at most 2. Let W be a subspace of P2
given by the span of {x2−x+6,−x2+2x−1,x+5}. Show that W is a proper
subspace of P2.

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